library(readr)
library(lmtest)
library(faraway)

Introduction

Conventional wisdom is that money buys happiness (winning) in Major League Baseball. However, the advent of “Moneyball” in the early 2000s by the Oakland Athletics, Cleveland Indians, and other teams, has lead to a more analytical approach to determining the make-up of Major League rosters.

As it turns out, money is not the magic elixir when it comes to assembling a winning Major League Baseball (MLB) team. The following plot shows that salary does not highly correlate with a winning record. This is substantiated by the companion single linear regression model and summary statistics that show that salary, while significant, only has a marginal impact on wins by an MLB team. The adjusted \(R^2\) from the simple linear regression – using training data from 2000 through 2013 – is low. Furthermore, the average percent error that compares actual wins versus predicted wins from the the test data (2014 through 2016) is high.

## 
## Call:
## lm(formula = W ~ salary, data = bbproj_trn)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -34.389  -8.180   0.701   8.404  35.659 
## 
## Coefficients:
##                   Estimate     Std. Error t value           Pr(>|t|)    
## (Intercept) 71.70962155199  1.29065984697  55.560            < 2e-16 ***
## salary       0.00000011551  0.00000001468   7.866 0.0000000000000316 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 10.86 on 418 degrees of freedom
## Multiple R-squared:  0.1289, Adjusted R-squared:  0.1269 
## F-statistic: 61.87 on 1 and 418 DF,  p-value: 0.00000000000003158
## [1] "Leave One Out Cross-Validated RMSE:  10.88"

So what are the factors that move the needle?

We explore two related threads in attempting to identify the factors that have a strong impact on a winning baseball team.

The first thread uses multiple linear regression to identify the factors that influence a team’s winning record over the course of a season. The second thread uses logistic regression to identify the factors that influence a team’s ability to win their division.

Data

A few words are in order about where we obtained the data from to perform this analysis.

The source of data is the Sean Lahman baseball archive (http://www.seanlahman.com/baseball-archive/), recognized by the Society for American Baseball Research (SABR) as the leading detailed player and team data archive from 1874 through the end of the 2017 Major League Baseball season. We use team statistics from the years 2000 through 2013 as the training dataset and use data from 2014 through 2016 as the test dataset. (As a note, we were unable to include the 2017 season in our analysis due to unavailability of salary data.)

Variables

The team-based variables we examine fall into one of three categories: offensive, defensive, or administrative.

Administrative

  • yearID - Year
  • franchID - Franchise, or team name
  • W - Wins (multiple linear regression response variable)
  • DivWin - Division winner (Y or N factor- Logistic linear regression response variable)
  • WCWin - Wild Card winner (Y or N factor)
  • LgWin - League champion (Y or N factor)
  • WSWin - World Series winner (Y or N factor)
  • salary - Team Salary (U.S dollars not adjusted for inflation)

Offensive

  • R - Runs scored
  • AB - At bats
  • H - Total hits (including doubles, triples, and home runs)
  • X1B - Singles
  • X2B - Doubles
  • X3B - Triples
  • HR - Home runs
  • BB - Base on balls (walks)
  • SO - Strikeouts
  • SB - Stolen bases
  • CS - Caught stealing
  • HBP - Hit by pitch
  • SF - Sacrifice flies
  • park - Name of team’s home ballpark (factor)
  • attendance - Home attendance total
  • BPF - Three-year park factor for batters
  • GIDP - Grounded into double plays
  • RBI - Runs Batted In
  • IBB - Intentional walks
  • TB - Total bases
  • SLG - Slugging percentage
  • OBP - On-base percentage
  • OPS - On-base plus slugging percentage
  • BABIP - Batting average for balls in play
  • RC - Runs created
  • uBB - Unintentional walks
  • wOBA - Weighted on-base average

Defensive

  • RA - Total runs allowed
  • ER - Earned runs allowed
  • ERA - Earned run average
  • CG - Complete games
  • SHO - Shutouts
  • SV - Saves
  • IPOuts - Outs pitched
  • HA - Hits allowed
  • HRA - Home runs allowed
  • BBA - Walks allowed
  • SOA - Strikeouts by pitchers
  • E - Errors
  • DP - Double plays
  • FP - Fielding percentage
  • PPF - Three-year park factor for pitchers
  • WHIP - Walks and hits per innings pitched

The Search for Better Multiple Linear Regression Models For Predicting Season Wins

Methodology

Better Candidate Multiple Linear Regression Models

scope <- W ~ R + AB + H + X2B + X3B + HR + BB + SO + SB + CS + HBP + SF + RA + 
  ER + ERA + CG + SHO + SV + IPouts + HA  + HRA + BBA + SOA + E + DP + FP + 
  attendance + BPF + PPF + salary + RBI + GIDP + IBB + TB + SLG + OBP + OPS + 
  WHIP + BABIP + RC + X1B + uBB + wOBA

formula <- formula(scope)
start_model <- lm(formula, data= bbproj_trn)
n <- length(resid(start_model))
step_search_start_model <- lm(W ~ 1, data= bbproj_trn)

Backwards Search: BIC Model

### Backwards Search: BIC Model
bic_model <- step(start_model,  direction = "backward", k = log(n), trace = 0)
summary(bic_model)
## 
## Call:
## lm(formula = W ~ R + AB + H + X2B + BB + SO + CS + SF + RA + 
##     CG + SHO + SV + IPouts + HRA + BBA + E + FP + GIDP + BABIP + 
##     RC, data = bbproj_trn)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -6.8573 -1.9757 -0.1555  1.9118  7.4927 
## 
## Coefficients:
##                 Estimate   Std. Error t value Pr(>|t|)    
## (Intercept)  1297.971349   339.322389   3.825 0.000152 ***
## R               0.076579     0.006140  12.472  < 2e-16 ***
## AB             -0.116786     0.014360  -8.133 5.37e-15 ***
## H               0.306262     0.049740   6.157 1.81e-09 ***
## X2B             0.034285     0.010085   3.399 0.000743 ***
## BB              0.031072     0.006669   4.659 4.33e-06 ***
## SO              0.051706     0.009680   5.341 1.55e-07 ***
## CS             -0.043025     0.015571  -2.763 0.005990 ** 
## SF             -0.095773     0.021708  -4.412 1.32e-05 ***
## RA             -0.054254     0.003811 -14.238  < 2e-16 ***
## CG              0.148895     0.046962   3.171 0.001639 ** 
## SHO             0.150078     0.049080   3.058 0.002380 ** 
## SV              0.345616     0.025203  13.714  < 2e-16 ***
## IPouts          0.057453     0.007126   8.063 8.78e-15 ***
## HRA            -0.026131     0.008530  -3.063 0.002338 ** 
## BBA            -0.008042     0.002619  -3.070 0.002287 ** 
## E              -0.182853     0.055690  -3.283 0.001116 ** 
## FP          -1045.006972   343.178691  -3.045 0.002480 ** 
## GIDP           -0.044072     0.012147  -3.628 0.000322 ***
## BABIP        -760.159504   137.620611  -5.524 6.00e-08 ***
## RC             -0.115957     0.024057  -4.820 2.04e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.729 on 399 degrees of freedom
## Multiple R-squared:  0.9475, Adjusted R-squared:  0.9449 
## F-statistic: 360.1 on 20 and 399 DF,  p-value: < 2.2e-16

Backwards Search: AIC Model

### Backwards Search: AIC Model
aic_model <- step(start_model, direction = "backward", trace = 0)
summary(aic_model)
## 
## Call:
## lm(formula = W ~ R + AB + H + X2B + BB + SO + CS + SF + RA + 
##     CG + SHO + SV + IPouts + HRA + BBA + E + FP + BPF + PPF + 
##     salary + GIDP + BABIP + RC, data = bbproj_trn)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -6.9901 -1.9391 -0.1523  1.9737  7.8582 
## 
## Coefficients:
##                      Estimate        Std. Error t value Pr(>|t|)    
## (Intercept) 1224.786161854707  345.895721393242   3.541 0.000446 ***
## R              0.080154881164    0.006191576342  12.946  < 2e-16 ***
## AB            -0.115370180878    0.014302511665  -8.066 8.71e-15 ***
## H              0.294198894471    0.049836065902   5.903 7.66e-09 ***
## X2B            0.034434010453    0.010055404520   3.424 0.000680 ***
## BB             0.029747234634    0.006800119630   4.375 1.56e-05 ***
## SO             0.048815441474    0.009683749780   5.041 7.06e-07 ***
## CS            -0.044627799778    0.015532577786  -2.873 0.004283 ** 
## SF            -0.096303670130    0.021621972055  -4.454 1.10e-05 ***
## RA            -0.057347471123    0.004031932044 -14.223  < 2e-16 ***
## CG             0.154701717514    0.046819638380   3.304 0.001039 ** 
## SHO            0.143225421132    0.048746440530   2.938 0.003495 ** 
## SV             0.346070608015    0.025076294754  13.801  < 2e-16 ***
## IPouts         0.058770788132    0.007084174725   8.296 1.71e-15 ***
## HRA           -0.022515047089    0.008581255343  -2.624 0.009033 ** 
## BBA           -0.008098302987    0.002628330947  -3.081 0.002206 ** 
## E             -0.172337651761    0.056648920630  -3.042 0.002505 ** 
## FP          -977.032306332758  348.883980530369  -2.800 0.005353 ** 
## BPF           -0.639264997534    0.233137357795  -2.742 0.006383 ** 
## PPF            0.599162430181    0.234965627610   2.550 0.011148 *  
## salary         0.000000008060    0.000000004559   1.768 0.077838 .  
## GIDP          -0.047142718419    0.012117297039  -3.891 0.000117 ***
## BABIP       -723.253375876375  138.139293383544  -5.236 2.67e-07 ***
## RC            -0.109319751926    0.024176237333  -4.522 8.11e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.704 on 396 degrees of freedom
## Multiple R-squared:  0.9489, Adjusted R-squared:  0.9459 
## F-statistic: 319.4 on 23 and 396 DF,  p-value: < 2.2e-16

Step Search: BIC Model

### Step Search: BIC Model
step_bic_model <- step(step_search_start_model, scope = scope, direction = "both",
                       k = log(n), trace = 0)
summary(step_bic_model)
## 
## Call:
## lm(formula = W ~ SV + RA + CG + OBP + X3B + SHO + R, data = bbproj_trn)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -9.7419 -1.9652  0.0052  1.8342  9.5000 
## 
## Coefficients:
##              Estimate Std. Error t value       Pr(>|t|)    
## (Intercept) 39.123870   6.025094   6.493 0.000000000242 ***
## SV           0.427352   0.025372  16.844        < 2e-16 ***
## RA          -0.076829   0.002616 -29.364        < 2e-16 ***
## CG           0.152982   0.046478   3.292       0.001082 ** 
## OBP         65.425764  23.940331   2.733       0.006549 ** 
## X3B         -0.063520   0.016876  -3.764       0.000192 ***
## SHO          0.174222   0.053140   3.279       0.001132 ** 
## R            0.079990   0.004115  19.439        < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.01 on 412 degrees of freedom
## Multiple R-squared:  0.9341, Adjusted R-squared:  0.9329 
## F-statistic: 833.7 on 7 and 412 DF,  p-value: < 2.2e-16

Step Search: AIC Model

### Step Search: AIC Model ### 
step_aic_model <- step(step_search_start_model, scope = scope, 
                       direction = "both", trace = 0)
summary(step_aic_model)
## 
## Call:
## lm(formula = W ~ SV + RA + CG + X3B + SHO + R + IPouts + AB + 
##     H + CS + GIDP + SF + BBA + HRA + E + FP + BPF + PPF + salary + 
##     HA, data = bbproj_trn)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -7.5195 -1.9115 -0.1021  1.8888  8.0195 
## 
## Coefficients:
##                      Estimate        Std. Error t value          Pr(>|t|)    
## (Intercept)  914.034003144572  347.045241981607   2.634          0.008773 ** 
## SV             0.355614506804    0.024640390657  14.432           < 2e-16 ***
## RA            -0.050247955833    0.006340159551  -7.925 0.000000000000023 ***
## CG             0.153017526099    0.044231471311   3.459          0.000600 ***
## X3B           -0.068574679286    0.016831959357  -4.074 0.000055748930342 ***
## SHO            0.138199572874    0.049132489791   2.813          0.005154 ** 
## R              0.084827624066    0.003584776861  23.663           < 2e-16 ***
## IPouts         0.061359854959    0.006880713970   8.918           < 2e-16 ***
## AB            -0.052857166713    0.005932828505  -8.909           < 2e-16 ***
## H              0.049033080922    0.006741047553   7.274 0.000000000001865 ***
## CS            -0.046638144061    0.015078914455  -3.093          0.002121 ** 
## GIDP          -0.043755357335    0.011669100346  -3.750          0.000203 ***
## SF            -0.052733758025    0.019580831875  -2.693          0.007377 ** 
## BBA           -0.010442337072    0.003185502948  -3.278          0.001137 ** 
## HRA           -0.021622522454    0.008707631554  -2.483          0.013432 *  
## E             -0.161327447529    0.057050198834  -2.828          0.004923 ** 
## FP          -884.643492393877  352.491658585801  -2.510          0.012479 *  
## BPF           -0.629501587254    0.230983379839  -2.725          0.006707 ** 
## PPF            0.584792736874    0.231600448463   2.525          0.011956 *  
## salary         0.000000007730    0.000000004537   1.704          0.089183 .  
## HA            -0.005934333997    0.004089404824  -1.451          0.147524    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.727 on 399 degrees of freedom
## Multiple R-squared:  0.9476, Adjusted R-squared:  0.945 
## F-statistic: 360.7 on 20 and 399 DF,  p-value: < 2.2e-16

Forward Search: BIC Model

bic_forward_model <- step(step_search_start_model, scope = scope, k = log(n),trace = 0)
summary(bic_forward_model)
## 
## Call:
## lm(formula = W ~ SV + RA + CG + OBP + X3B + SHO + R, data = bbproj_trn)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -9.7419 -1.9652  0.0052  1.8342  9.5000 
## 
## Coefficients:
##              Estimate Std. Error t value       Pr(>|t|)    
## (Intercept) 39.123870   6.025094   6.493 0.000000000242 ***
## SV           0.427352   0.025372  16.844        < 2e-16 ***
## RA          -0.076829   0.002616 -29.364        < 2e-16 ***
## CG           0.152982   0.046478   3.292       0.001082 ** 
## OBP         65.425764  23.940331   2.733       0.006549 ** 
## X3B         -0.063520   0.016876  -3.764       0.000192 ***
## SHO          0.174222   0.053140   3.279       0.001132 ** 
## R            0.079990   0.004115  19.439        < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.01 on 412 degrees of freedom
## Multiple R-squared:  0.9341, Adjusted R-squared:  0.9329 
## F-statistic: 833.7 on 7 and 412 DF,  p-value: < 2.2e-16

Forward Search: AIC Model

aic_forward_model <-  step(step_search_start_model, scope = scope, trace = 0)
summary(aic_forward_model)
## 
## Call:
## lm(formula = W ~ SV + RA + CG + X3B + SHO + R + IPouts + AB + 
##     H + CS + GIDP + SF + BBA + HRA + E + FP + BPF + PPF + salary + 
##     HA, data = bbproj_trn)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -7.5195 -1.9115 -0.1021  1.8888  8.0195 
## 
## Coefficients:
##                      Estimate        Std. Error t value          Pr(>|t|)    
## (Intercept)  914.034003144572  347.045241981607   2.634          0.008773 ** 
## SV             0.355614506804    0.024640390657  14.432           < 2e-16 ***
## RA            -0.050247955833    0.006340159551  -7.925 0.000000000000023 ***
## CG             0.153017526099    0.044231471311   3.459          0.000600 ***
## X3B           -0.068574679286    0.016831959357  -4.074 0.000055748930342 ***
## SHO            0.138199572874    0.049132489791   2.813          0.005154 ** 
## R              0.084827624066    0.003584776861  23.663           < 2e-16 ***
## IPouts         0.061359854959    0.006880713970   8.918           < 2e-16 ***
## AB            -0.052857166713    0.005932828505  -8.909           < 2e-16 ***
## H              0.049033080922    0.006741047553   7.274 0.000000000001865 ***
## CS            -0.046638144061    0.015078914455  -3.093          0.002121 ** 
## GIDP          -0.043755357335    0.011669100346  -3.750          0.000203 ***
## SF            -0.052733758025    0.019580831875  -2.693          0.007377 ** 
## BBA           -0.010442337072    0.003185502948  -3.278          0.001137 ** 
## HRA           -0.021622522454    0.008707631554  -2.483          0.013432 *  
## E             -0.161327447529    0.057050198834  -2.828          0.004923 ** 
## FP          -884.643492393877  352.491658585801  -2.510          0.012479 *  
## BPF           -0.629501587254    0.230983379839  -2.725          0.006707 ** 
## PPF            0.584792736874    0.231600448463   2.525          0.011956 *  
## salary         0.000000007730    0.000000004537   1.704          0.089183 .  
## HA            -0.005934333997    0.004089404824  -1.451          0.147524    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.727 on 399 degrees of freedom
## Multiple R-squared:  0.9476, Adjusted R-squared:  0.945 
## F-statistic: 360.7 on 20 and 399 DF,  p-value: < 2.2e-16

Evaluation and Refinement of the Candidate Models

Evaluation of the Backwards Search: BIC Model

### Breusch-Pagan Test on Backwards Search - BIC Model
library(lmtest)
bptest(bic_model)
## 
##  studentized Breusch-Pagan test
## 
## data:  bic_model
## BP = 19.912, df = 20, p-value = 0.4635
plot_fitted_versus_residuals(fitted(bic_model), 
                             resid(bic_model), 
                             "Backwards Search - BIC Model")

### Shapiro - Wilk Normality Test on Backwards Search - BIC Model ###
shapiro.test(resid(bic_model))
## 
##  Shapiro-Wilk normality test
## 
## data:  resid(bic_model)
## W = 0.99328, p-value = 0.05825
qqnorm(resid(bic_model), 
       main = "Normal Q-Q Plot, Backwards Search - BIC Model", 
       col = "darkgrey")
qqline(resid(bic_model), col = "dodgerblue", lwd = 2)

print(paste("Leave One Out Cross-Validated RMSE: ", 
            round(calc_loocv_rmse(bic_model), 2)))
## [1] "Leave One Out Cross-Validated RMSE:  2.81"
### Do the number of standard residuals greater than 2 exceed 5% of the total 
### observations -- Backwards Search: BIC Model
std_resid_bic_model <- rstandard(bic_model)[abs(rstandard(bic_model)) > 2]
is_std_resid_gt_five_percent_bic_model <- length(std_resid_bic_model) / n > 0.05
ifelse(is_std_resid_gt_five_percent_bic_model, 
       "Outliers Exceed 5% of Obs", 
       "Outliers Do Not Exceed 5% of Obs")
## [1] "Outliers Do Not Exceed 5% of Obs"
### VIF > 5 for Backwards Search: BIC Model Coefficients
vif_bic_model <- vif(bic_model)
vif_bic_model[which(vif_bic_model > 5)]
##          R         AB          H         BB         SO         RA          E         FP      BABIP         RC 
##  14.894205  72.403111 929.925966  12.247576  80.989904   6.226679  48.143435  48.871767 122.560946 226.450179

Refining the Backwards Search: BIC Model Due to High Variance Inflation Factors

library(caret)
bic_model_high_vif_cols <- c("R", "AB", "H", "BB", "SO", "RA", "E", "FP",
                             "BABIP", "RC")
indices_to_drop <- findCorrelation(cor(bbproj_trn[,c(bic_model_high_vif_cols)]), 
                                   cutoff = 0.6)
(vars_to_drop <- bic_model_high_vif_cols[indices_to_drop])
## [1] "H"  "RC" "R"  "FP"
A Better And Smaller Backwards Search: BIC Model
smaller_bic_model <-  lm(W ~ AB  + BB + SO + SF + RA + CG + SV + IPouts +  BBA + BABIP, data = bbproj_trn)
summary(smaller_bic_model)
## 
## Call:
## lm(formula = W ~ AB + BB + SO + SF + RA + CG + SV + IPouts + 
##     BBA + BABIP, data = bbproj_trn)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -13.0090  -3.1465   0.0746   3.1017  13.2658 
## 
## Coefficients:
##               Estimate Std. Error t value       Pr(>|t|)    
## (Intercept) -61.891791  33.514874  -1.847       0.065513 .  
## AB            0.035857   0.005639   6.358 0.000000000546 ***
## BB            0.055611   0.003754  14.812        < 2e-16 ***
## SO           -0.008264   0.002508  -3.296       0.001067 ** 
## SF            0.083700   0.033456   2.502       0.012746 *  
## RA           -0.070294   0.004609 -15.250        < 2e-16 ***
## CG            0.335688   0.080389   4.176 0.000036315586 ***
## SV            0.599497   0.041742  14.362        < 2e-16 ***
## IPouts       -0.019466   0.010220  -1.905       0.057516 .  
## BBA          -0.008600   0.004530  -1.898       0.058341 .  
## BABIP       118.264318  33.616837   3.518       0.000484 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.888 on 409 degrees of freedom
## Multiple R-squared:  0.8274, Adjusted R-squared:  0.8231 
## F-statistic:   196 on 10 and 409 DF,  p-value: < 2.2e-16
Diagnostics for the Better and Smaller Backwards Search: BIC Model
model_diagnostics(smaller_bic_model)
## [1] "Variance Inflation Factors"
##       AB       BB       SO       SF       RA       CG       SV   IPouts      BBA    BABIP 
## 3.480184 1.210000 1.693955 1.472823 2.839735 1.243693 1.573034 2.843318 1.558630 2.279289 
## [1] "Number of coefficients with VIF > 5:  0"
## 
##  studentized Breusch-Pagan test
## 
## data:  model
## BP = 13.488, df = 10, p-value = 0.1976
## 
## 
##  Shapiro-Wilk normality test
## 
## data:  resid(model)
## W = 0.99706, p-value = 0.6567
## 
## [1] "Leave One Out Cross-Validated RMSE:  4.95"
## [1] "Outliers Exceed 5% of Obs"

Evaluation of the Backwards Search: AIC Model

### Breusch-Pagan Test on AIC Model
bptest(aic_model)
## 
##  studentized Breusch-Pagan test
## 
## data:  aic_model
## BP = 22.713, df = 23, p-value = 0.4777
plot_fitted_versus_residuals(fitted(aic_model), resid(aic_model), 
                             "Backwards Search - AIC Model")

### Shapiro - Wilk Normality Test on Backwards Search - AIC Model ###
shapiro.test(resid(aic_model))
## 
##  Shapiro-Wilk normality test
## 
## data:  resid(aic_model)
## W = 0.99355, p-value = 0.07005
qqnorm(resid(aic_model), 
       main = "Normal Q-Q Plot, Backwards Search - AIC Model", 
       col = "darkgrey")
qqline(resid(aic_model), col = "dodgerblue", lwd = 2)

## [1] "Leave One Out Cross-Validated RMSE:  2.79"
### Do the number of standard residuals greater than 2 exceed 5% of the total 
### observations -- Backwards Search: AIC Model
std_resid_aic_model <- rstandard(aic_model)[abs(rstandard(aic_model)) > 2]
is_std_resid_gt_five_percent_aic_model <- length(std_resid_aic_model) / n > 0.05
ifelse(is_std_resid_gt_five_percent_aic_model, 
       "Outliers Exceed 5% of Obs", 
       "Outliers Do Not Exceed 5% of Obs")
## [1] "Outliers Do Not Exceed 5% of Obs"
### VIF > 5 for Backwards Search: AIC Model Coefficients
vif_aic_model <- vif(aic_model)
vif_aic_model[which(vif_aic_model > 5)]
##          R         AB          H         BB         SO         RA          E         FP        BPF        PPF      BABIP         RC 
##  15.427663  73.154978 950.861657  12.972390  82.555959   7.100542  50.740746  51.448558  85.281044  84.114946 125.780508 232.943354

Refining the Backwards Search: AIC Model Due to High Variance Inflation Factors

aic_model_high_vif_cols <- c("R", "AB", "H", "X2B", "X3B", "HR", "BB", "SO", 
                             "SF", "RA", "ER", "IPouts", "E", "FP","BPF", "PPF", 
                             "IBB", "OBP", "BABIP", "RC", "wOBA")
indices_to_drop <- findCorrelation(cor(bbproj_trn[,c(aic_model_high_vif_cols)]), 
                                   cutoff = 0.6)
(vars_to_drop <- aic_model_high_vif_cols[indices_to_drop])
## [1] "RC"   "wOBA" "R"    "OBP"  "H"    "RA"   "BPF"  "FP"
A Better And Smaller Backwards Search: AIC Model
# smaller_aic_model <-  lm(formula = W ~  AB + HR + BB + SO + CS + ER + CG + 
#                            SHO + SV + IPouts + BBA + FP + GIDP + IBB +  BABIP, 
#                          data = bbproj)
smaller_aic_model <-  lm(formula = W ~  HR + BB + SO + ER + CG + SHO + SV + 
                           IPouts + BBA + FP + GIDP + BABIP, data = bbproj_trn)
summary(smaller_aic_model)
## 
## Call:
## lm(formula = W ~ HR + BB + SO + ER + CG + SHO + SV + IPouts + 
##     BBA + FP + GIDP + BABIP, data = bbproj_trn)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -8.3371 -2.5236 -0.0841  2.1182 10.2418 
## 
## Coefficients:
##                Estimate  Std. Error t value   Pr(>|t|)    
## (Intercept) -328.428863   67.755211  -4.847 0.00000178 ***
## HR             0.132223    0.005998  22.044    < 2e-16 ***
## BB             0.032238    0.002798  11.523    < 2e-16 ***
## SO            -0.023360    0.001674 -13.952    < 2e-16 ***
## ER            -0.068993    0.003633 -18.993    < 2e-16 ***
## CG             0.189266    0.057990   3.264   0.001192 ** 
## SHO            0.235242    0.061006   3.856   0.000134 ***
## SV             0.410129    0.030171  13.593    < 2e-16 ***
## IPouts         0.014815    0.005185   2.857   0.004495 ** 
## BBA           -0.008063    0.003203  -2.517   0.012216 *  
## FP           278.765319   69.652956   4.002 0.00007457 ***
## GIDP          -0.037475    0.012793  -2.929   0.003589 ** 
## BABIP        316.088714   16.896493  18.707    < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.435 on 407 degrees of freedom
## Multiple R-squared:  0.9152, Adjusted R-squared:  0.9127 
## F-statistic: 365.9 on 12 and 407 DF,  p-value: < 2.2e-16
Diagnostics for the Better and Smaller Backwards Search: AIC Model
model_diagnostics(smaller_aic_model)
## [1] "Variance Inflation Factors"
##       HR       BB       SO       ER       CG      SHO       SV   IPouts      BBA       FP     GIDP    BABIP 
## 1.426241 1.360620 1.529067 3.064298 1.310541 1.907533 1.664221 1.482259 1.578259 1.270639 1.409269 1.166015 
## [1] "Number of coefficients with VIF > 5:  0"
## 
##  studentized Breusch-Pagan test
## 
## data:  model
## BP = 8.7708, df = 12, p-value = 0.7224
## 
## 
##  Shapiro-Wilk normality test
## 
## data:  resid(model)
## W = 0.99241, p-value = 0.03137
## 
## [1] "Leave One Out Cross-Validated RMSE:  3.49"
## [1] "Outliers Exceed 5% of Obs"

Evaluation of the Step Search: BIC Model

### Breusch-Pagan Test on Step BIC Model
bptest(step_bic_model)
## 
##  studentized Breusch-Pagan test
## 
## data:  step_bic_model
## BP = 3.0387, df = 7, p-value = 0.8814
plot_fitted_versus_residuals(fitted(step_bic_model), resid(step_bic_model), 
                             "Step Search - BIC Model")

### Shapiro - Wilk Normality Test on Step Search - BIC Model ###
shapiro.test(resid(step_bic_model))
## 
##  Shapiro-Wilk normality test
## 
## data:  resid(step_bic_model)
## W = 0.99708, p-value = 0.6604
qqnorm(resid(step_bic_model), 
       main = "Normal Q-Q Plot, Step Search - BIC Model", 
       col = "darkgrey")
qqline(resid(step_bic_model), col = "dodgerblue", lwd = 2)

## [1] "Leave One Out Cross-Validated RMSE:  3.04"
### Do the number of standard residuals greater than 2 exceed 5% of the total
### observations -- Step Search: BIC Model
std_resid_step_bic_model <- rstandard(
  step_bic_model)[abs(rstandard(step_bic_model)) > 2]
is_std_resid_gt_five_percent__step_bic_model <- length(
  std_resid_step_bic_model) / n > 0.05
ifelse(is_std_resid_gt_five_percent__step_bic_model, "Exceeds 5% of Obs", 
       "Does Not Exceed 5% of Obs")
## [1] "Does Not Exceed 5% of Obs"
### VIF > 5 for Step Search: BIC Model Coefficients
vif_step_bic_model <- vif(step_bic_model)
vif_step_bic_model[which(vif_step_bic_model > 5)]
##      OBP        R 
## 5.221787 5.499516

Refining the Step Search: BIC Model Due to High Variance Inflation Factors

A Better And Smaller Step Search: BIC Model
smaller_step_bic_model <- lm(formula = W ~ SV + R + RA + SHO + CG + X3B + 
                               IPouts + AB + salary, data = bbproj_trn)
summary(smaller_step_bic_model)
## 
## Call:
## lm(formula = W ~ SV + R + RA + SHO + CG + X3B + IPouts + AB + 
##     salary, data = bbproj_trn)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -8.6930 -2.0156  0.0027  1.8031  8.6432 
## 
## Coefficients:
##                   Estimate     Std. Error t value  Pr(>|t|)    
## (Intercept) 16.64791877157 19.43171439428   0.857   0.39209    
## SV           0.39794921940  0.02548723971  15.614   < 2e-16 ***
## R            0.09465788997  0.00239639476  39.500   < 2e-16 ***
## RA          -0.07000030397  0.00295754734 -23.668   < 2e-16 ***
## SHO          0.17217667234  0.05222372960   3.297   0.00106 ** 
## CG           0.14266098090  0.04629864831   3.081   0.00220 ** 
## X3B         -0.04901240782  0.01691431490  -2.898   0.00396 ** 
## IPouts       0.02307347246  0.00528945937   4.362 0.0000163 ***
## AB          -0.01298730710  0.00293489287  -4.425 0.0000124 ***
## salary       0.00000001035  0.00000000449   2.305   0.02167 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.95 on 410 degrees of freedom
## Multiple R-squared:  0.937,  Adjusted R-squared:  0.9356 
## F-statistic: 677.2 on 9 and 410 DF,  p-value: < 2.2e-16
Diagnostics for the Better and Smaller Step Search: BIC Model
model_diagnostics(smaller_step_bic_model)
## [1] "Variance Inflation Factors"
##       SV        R       RA      SHO       CG      X3B   IPouts       AB   salary 
## 1.610244 1.941649 3.209850 1.895339 1.132668 1.067672 2.091258 2.587975 1.267406 
## [1] "Number of coefficients with VIF > 5:  0"
## 
##  studentized Breusch-Pagan test
## 
## data:  model
## BP = 4.372, df = 9, p-value = 0.8853
## 
## 
##  Shapiro-Wilk normality test
## 
## data:  resid(model)
## W = 0.9967, p-value = 0.549
## 
## [1] "Leave One Out Cross-Validated RMSE:  2.99"
## [1] "Outliers Do Not Exceed 5% of Obs"

Evaluation of the Step Search: AIC Model

### Breusch-Pagan Test on Step AIC Model
bptest(step_aic_model)
## 
##  studentized Breusch-Pagan test
## 
## data:  step_aic_model
## BP = 22.885, df = 20, p-value = 0.2945
plot_fitted_versus_residuals(fitted(step_aic_model), resid(step_aic_model), 
                             "Step Search - AIC Model")

### Shapiro - Wilk Normality Test on Step Search - AIC Model ###
shapiro.test(resid(step_aic_model))
## 
##  Shapiro-Wilk normality test
## 
## data:  resid(step_aic_model)
## W = 0.99683, p-value = 0.5883
qqnorm(resid(step_aic_model), 
       main = "Normal Q-Q Plot, Step Search - AIC Model", 
       col = "darkgrey")
qqline(resid(step_aic_model), col = "dodgerblue", lwd = 2)

## [1] "Leave One Out Cross-Validated RMSE:  2.8"
### Do the number of standard residuals greater than 2 exceed 5% of the total
### observations -- Step Search: AIC Model
std_resid_step_aic_model <- rstandard(
  step_aic_model)[abs(rstandard(step_aic_model)) > 2]
is_std_resid_gt_five_percent__step_aic_model <- length(
  std_resid_step_aic_model) / n > 0.05
ifelse(is_std_resid_gt_five_percent__step_aic_model,
       "Exceeds 5% of Obs", 
       "Does Not Exceed 5% of Obs")
## [1] "Does Not Exceed 5% of Obs"
### VIF > 5 for Step Search: AIC Model Coefficients
vif_step_aic_model <- vif(step_aic_model)
vif_step_aic_model[which(vif_step_aic_model > 5)]
##        RA         R        AB         H         E        FP       BPF       PPF        HA 
## 17.262491  5.084637 12.376016 17.104981 50.597130 51.635315 82.305380 80.349149  7.123143

Refining the Step Search: AIC Model Due to High Variance Inflation Factors

A Better And Smaller Step Search: AIC Model
smaller_step_aic_model <- lm(formula = W ~ SV + R + SHO + CG + X3B + IPouts + 
                               BBA + HRA + SOA, data = bbproj_trn)
summary(smaller_step_aic_model)
## 
## Call:
## lm(formula = W ~ SV + R + SHO + CG + X3B + IPouts + BBA + HRA + 
##     SOA, data = bbproj_trn)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -11.9548  -2.9275   0.2111   2.5960  15.3693 
## 
## Coefficients:
##                Estimate  Std. Error t value          Pr(>|t|)    
## (Intercept) -106.910065   24.782771  -4.314 0.000020123812728 ***
## SV             0.553970    0.033062  16.756           < 2e-16 ***
## R              0.082172    0.002520  32.612           < 2e-16 ***
## SHO            0.459052    0.068216   6.729 0.000000000057562 ***
## CG             0.295343    0.063584   4.645 0.000004589228066 ***
## X3B           -0.063341    0.022615  -2.801           0.00534 ** 
## IPouts         0.026697    0.005825   4.583 0.000006087870449 ***
## BBA           -0.027269    0.003450  -7.905 0.000000000000025 ***
## HRA           -0.098460    0.009868  -9.978           < 2e-16 ***
## SOA            0.013936    0.001999   6.970 0.000000000012736 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.006 on 410 degrees of freedom
## Multiple R-squared:  0.8837, Adjusted R-squared:  0.8812 
## F-statistic: 346.3 on 9 and 410 DF,  p-value: < 2.2e-16
Diagnostics for the Better and Smaller Step Search: AIC Model
model_diagnostics(smaller_step_aic_model)
## [1] "Variance Inflation Factors"
##       SV        R      SHO       CG      X3B   IPouts      BBA      HRA      SOA 
## 1.468999 1.163807 1.753261 1.158178 1.034795 1.375007 1.345455 1.577315 1.440348 
## [1] "Number of coefficients with VIF > 5:  0"
## 
##  studentized Breusch-Pagan test
## 
## data:  model
## BP = 3.5705, df = 9, p-value = 0.9373
## 
## 
##  Shapiro-Wilk normality test
## 
## data:  resid(model)
## W = 0.99481, p-value = 0.1694
## 
## [1] "Leave One Out Cross-Validated RMSE:  4.05"
## [1] "Outliers Do Not Exceed 5% of Obs"

Evaluation of the Forward Search: BIC Model

### Breusch-Pagan Test
bptest(bic_forward_model)
## 
##  studentized Breusch-Pagan test
## 
## data:  bic_forward_model
## BP = 3.0387, df = 7, p-value = 0.8814
plot_fitted_versus_residuals(fitted(bic_forward_model), resid(bic_forward_model), "Forward Search - BIC Model")

### Shapiro - Wilk Normality Test ###
shapiro.test(resid(bic_forward_model))
## 
##  Shapiro-Wilk normality test
## 
## data:  resid(bic_forward_model)
## W = 0.99708, p-value = 0.6604
qqnorm(resid(bic_forward_model), main = "Normal Q-Q Plot, Step Search - BIC Model", col = "darkgrey")
qqline(resid(bic_forward_model), col = "dodgerblue", lwd = 2)

## [1] "Leave One Out Cross-Validated RMSE:  3.04"
### Do the number of standard residuals greater than 2 exceed 5% of the total observations -- Step Search: AIC Model
std_resid <- rstandard(bic_forward_model)[abs(rstandard(bic_forward_model)) > 2]
is_std_resid_gt_five_percent <- length(std_resid) / n > 0.05
ifelse(is_std_resid_gt_five_percent,"Exceeds 5% of Obs", "Does Not Exceed 5% of Obs")
## [1] "Does Not Exceed 5% of Obs"
### VIF > 5 for Step Search: AIC Model Coefficients
library(faraway)
vifs <- vif(bic_forward_model)
vifs[which(vifs > 5)]
##      OBP        R 
## 5.221787 5.499516

Refining the Forward Search: BIC Model Due to High Variance Inflation Factors

library(caret)
high_vif_cols <- as.array(names(vifs[which(vifs > 5)]))
indices_to_drop <- findCorrelation(cor(bbproj_trn[,c(high_vif_cols)]), cutoff = 0.6)
vars_to_drop <- high_vif_cols[indices_to_drop]
vars_to_drop
## [1] "R"
A Better And Smaller Forward Search: BIC Model
terms = attr(bic_forward_model$terms, "term.labels")
terms = terms[! terms %in% vars_to_drop]
formula = as.formula(paste("W ~", paste(terms, collapse = "+")))
smaller_model <-  lm(formula = formula, data = bbproj_trn)
summary(smaller_model)
## 
## Call:
## lm(formula = formula, data = bbproj_trn)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -11.2187  -2.7593  -0.0139   2.6440  11.9379 
## 
## Coefficients:
##               Estimate Std. Error t value  Pr(>|t|)    
## (Intercept) -48.379453   5.538267  -8.735   < 2e-16 ***
## SV            0.510814   0.034581  14.771   < 2e-16 ***
## RA           -0.067633   0.003559 -19.005   < 2e-16 ***
## CG            0.259678   0.063826   4.069 0.0000567 ***
## OBP         478.519624  15.245069  31.388   < 2e-16 ***
## X3B          -0.058658   0.023336  -2.514    0.0123 *  
## SHO           0.166892   0.073487   2.271    0.0237 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.162 on 413 degrees of freedom
## Multiple R-squared:  0.8736, Adjusted R-squared:  0.8717 
## F-statistic: 475.7 on 6 and 413 DF,  p-value: < 2.2e-16
smaller_bic_forward_model = smaller_model
Diagnostics for the Better and Smaller Forward Search: BIC Model
model_diagnostics(smaller_bic_forward_model)
## [1] "Variance Inflation Factors"
##       SV       RA       CG      OBP      X3B      SHO 
## 1.488879 2.334176 1.081155 1.107182 1.020757 1.884954 
## [1] "Number of coefficients with VIF > 5:  0"
## 
##  studentized Breusch-Pagan test
## 
## data:  model
## BP = 3.5489, df = 6, p-value = 0.7375
## 
## 
##  Shapiro-Wilk normality test
## 
## data:  resid(model)
## W = 0.99807, p-value = 0.9205
## 
## [1] "Leave One Out Cross-Validated RMSE:  4.2"
## [1] "Outliers Do Not Exceed 5% of Obs"

Evaluation of the Forward Search: AIC Model

### Breusch-Pagan Test
bptest(aic_forward_model)
## 
##  studentized Breusch-Pagan test
## 
## data:  aic_forward_model
## BP = 22.885, df = 20, p-value = 0.2945
plot_fitted_versus_residuals(fitted(aic_forward_model), resid(aic_forward_model), "Forward Search - AIC Model")

### Shapiro - Wilk Normality Test ###
shapiro.test(resid(aic_forward_model))
## 
##  Shapiro-Wilk normality test
## 
## data:  resid(aic_forward_model)
## W = 0.99683, p-value = 0.5883
qqnorm(resid(aic_forward_model), main = "Normal Q-Q Plot, Step Search - AIC Model", col = "darkgrey")
qqline(resid(aic_forward_model), col = "dodgerblue", lwd = 2)

## [1] "Leave One Out Cross-Validated RMSE:  2.8"
### Do the number of standard residuals greater than 2 exceed 5% of the total observations -- Step Search: AIC Model
std_resid <- rstandard(aic_forward_model)[abs(rstandard(aic_forward_model)) > 2]
is_std_resid_gt_five_percent <- length(std_resid) / n > 0.05
ifelse(is_std_resid_gt_five_percent,"Exceeds 5% of Obs", "Does Not Exceed 5% of Obs")
## [1] "Does Not Exceed 5% of Obs"
### VIF > 5 for Step Search: AIC Model Coefficients
library(faraway)
vifs <- vif(aic_forward_model)
vifs[which(vifs > 5)]
##        RA         R        AB         H         E        FP       BPF       PPF        HA 
## 17.262491  5.084637 12.376016 17.104981 50.597130 51.635315 82.305380 80.349149  7.123143

Refining the Forward Search: AIC Model Due to High Variance Inflation Factors

library(caret)
high_vif_cols <- as.array(names(vifs[which(vifs > 5)]))
indices_to_drop <- findCorrelation(cor(bbproj_trn[,c(high_vif_cols)]), cutoff = 0.6)
vars_to_drop <- high_vif_cols[indices_to_drop]
vars_to_drop
## [1] "H"   "RA"  "R"   "BPF" "FP"
A Better And Smaller Forward Search: AIC Model
terms = attr(aic_forward_model$terms, "term.labels")
terms = terms[! terms %in% vars_to_drop]
formula = as.formula(paste("W ~", paste(terms, collapse = "+")))
smaller_model <-  lm(formula = formula, data = bbproj_trn)
summary(smaller_model)
## 
## Call:
## lm(formula = formula, data = bbproj_trn)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -20.7707  -4.0132   0.0735   3.9381  16.8375 
## 
## Coefficients:
##                      Estimate        Std. Error t value         Pr(>|t|)    
## (Intercept) -130.372691160134   40.942184890385  -3.184         0.001563 ** 
## SV             0.691822762359    0.051293290999  13.488          < 2e-16 ***
## CG             0.451451517680    0.097869852286   4.613 0.00000534151951 ***
## X3B           -0.044130287173    0.037206097668  -1.186         0.236278    
## SHO            0.187051414650    0.110061882108   1.700         0.089993 .  
## IPouts         0.006637885812    0.010439115229   0.636         0.525223    
## AB             0.035276686983    0.005368182209   6.571 0.00000000015396 ***
## CS            -0.018371247437    0.029837527171  -0.616         0.538433    
## GIDP           0.026472866957    0.020953425350   1.263         0.207169    
## SF             0.273938786852    0.038676525292   7.083 0.00000000000631 ***
## BBA           -0.020850806318    0.005466335198  -3.814         0.000158 ***
## HRA           -0.033796874832    0.016267982447  -2.078         0.038386 *  
## E             -0.045858530274    0.020678376355  -2.218         0.027131 *  
## PPF            0.154768090993    0.063792547637   2.426         0.015698 *  
## salary         0.000000022080    0.000000009774   2.259         0.024405 *  
## HA            -0.037682947274    0.005416312830  -6.957 0.00000000001406 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 6.152 on 404 degrees of freedom
## Multiple R-squared:  0.7299, Adjusted R-squared:  0.7198 
## F-statistic: 72.77 on 15 and 404 DF,  p-value: < 2.2e-16
smaller_aic_forward_model = smaller_model
Diagnostics for the Better and Smaller Forward Search: AIC Model
model_diagnostics(smaller_aic_forward_model)
## [1] "Variance Inflation Factors"
##       SV       CG      X3B      SHO   IPouts       AB       CS     GIDP       SF      BBA      HRA        E      PPF   salary 
## 1.499417 1.163642 1.187716 1.935442 1.872698 1.990606 1.231270 1.178478 1.242515 1.432725 1.817827 1.305924 1.197612 1.380530 
##       HA 
## 2.454893 
## [1] "Number of coefficients with VIF > 5:  0"
## 
##  studentized Breusch-Pagan test
## 
## data:  model
## BP = 20.617, df = 15, p-value = 0.1495
## 
## 
##  Shapiro-Wilk normality test
## 
## data:  resid(model)
## W = 0.9972, p-value = 0.6962
## 
## [1] "Leave One Out Cross-Validated RMSE:  6.27"
## [1] "Outliers Do Not Exceed 5% of Obs"

Validate Model Effectiveness Using Test Data

Results

And the Winning Linear Regression Model is…

Discussion


Logistic Modeling and Prediction of Division Winners

Salary as a Predictor for Division Winners

Let’s turn our attention to logistic regression and our ability to classify and predict division winners using our team predictor set. First, let’s see how well salary alone can predict pennants.

salary_only_model <- glm(DivWin ~ salary, data = bbproj_trn, family = binomial)
(salary_only_summary = summary(salary_only_model))
## 
## Call:
## glm(formula = DivWin ~ salary, family = binomial, data = bbproj_trn)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -1.5538  -0.6746  -0.5664  -0.4673   2.1545  
## 
## Coefficients:
##                    Estimate      Std. Error z value   Pr(>|z|)    
## (Intercept) -2.693550028350  0.317777033524  -8.476    < 2e-16 ***
## salary       0.000000015283  0.000000003249   4.704 0.00000255 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 420.34  on 419  degrees of freedom
## Residual deviance: 397.38  on 418  degrees of freedom
## AIC: 401.38
## 
## Number of Fisher Scoring iterations: 4
plot(as.numeric(DivWin) - 1 ~ salary, data = bbproj_trn, 
     pch = 20, 
     main = "Probability of Buying a Division Winner",
     ylab = "Probability of Winning Division", 
     xlab = "Team Salary ($)",  
     xlim = c(0, 3e8))
curve(predict(salary_only_model, data.frame(salary = x), type = "response"), 
      add = TRUE, 
      col = "tomato", 
      lty = 2,
      lwd = 2)

Graphically, a team’s payroll seems to be moderately influential in predicting their chances of taking home a division crown. Indeed, our Wald test for salary alone yields a p-value of 0.0000025, allowing us to reject the null hypothesis (\(H_0 : \beta_{salary} = 0\)) for any reasonable value for \(\alpha\). So what happens when we look at our salary model’s misclassification rate?

salary_prediction <- ifelse(predict(salary_only_model, 
                                    bbproj_tst, 
                                    type = "response") > 0.5, "Y", "N")
(prevalence = table(bbproj_tst$DivWin) / nrow(bbproj_tst))
## 
##   N   Y 
## 0.8 0.2
(salary_misclass = mean(salary_prediction != bbproj_tst$DivWin))
## [1] 0.2222222

First, let’s examine prevalence of division winners. We see that 20% of the teams were division winners, which makes sense, since there are 6 divisions and 30 MLB teams, therefore you will only have 6 division winners per year. Our salary model has a misclassification rate of 0.222, which is worse than our prevalence. We would have a better misclassification rate if we simply stated that there are no division winners! We can certainly do better than this.

Methodology

Candidate Logistic Regression Models

We will begin our search for a better classifier by setting up an initial logistic regression model contain all of the predictors we would like to evaluate. Then, we will proceed to eliminate predictors using backwards, forwards, and stepwise AIC and BIC searches. See Appendix C for a complete evaluation of all of the models. The following will detail the methodology using our best candidate model.

scope <- DivWin ~ W + R + AB + H + X2B + X3B + HR + BB + SO + SB + CS + HBP + 
  SF + RA + ER + ERA + CG + SHO + SV + IPouts + HA  + HRA + BBA + SOA + E + 
  DP + FP + attendance + BPF + PPF + salary + RBI + GIDP + IBB + TB + SLG + 
  OBP + OPS + WHIP + BABIP + RC + X1B + uBB + wOBA
start_model <- glm(DivWin ~ 1, data = bbproj_trn, family = binomial)
n <- length(resid(start_model))

Forward Search: BIC Model

bic_model <- step(start_model, direction = "forward", scope = scope, 
                  k = log(n), trace = 0)
summary(bic_model)
## 
## Call:
## glm(formula = DivWin ~ W + X2B, family = binomial, data = bbproj_trn)
## 
## Deviance Residuals: 
##      Min        1Q    Median        3Q       Max  
## -2.35987  -0.30139  -0.06081  -0.00867   2.59348  
## 
## Coefficients:
##               Estimate Std. Error z value          Pr(>|z|)    
## (Intercept) -27.523331   3.832156  -7.182 0.000000000000686 ***
## W             0.350030   0.042623   8.212           < 2e-16 ***
## X2B          -0.016344   0.006758  -2.418            0.0156 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 420.34  on 419  degrees of freedom
## Residual deviance: 194.55  on 417  degrees of freedom
## AIC: 200.55
## 
## Number of Fisher Scoring iterations: 7

Our forward search using BIC yields the following logistic model

\[ \log \bigg(\frac{P[DivWin = 1]}{1 - P[DivWin = 1]} \bigg) = \beta_0 + \beta_W W + \beta_{X2B} X2B \]

where the log odds of a team winning their division is dependent upon wins and doubles.

Diagnostics: BIC Model

Next, we will create a function that will evaluate this model using a confusion matrix and calculate its sensitivity, specificity, and its misclassification rate (the classify_and_diagnose function code can be found in Appendix C). Then we will start by examining a default cutoff value of 0.5.

(bic_diag <- classify_and_diagnose(bic_model))
## $confusion_matrix
##           actual
## prediction  N  Y
##          N 70  7
##          Y  2 11
## 
## $sensitivity
## [1] 0.6111111
## 
## $specificity
## [1] 0.9722222
## 
## $misclassification
## [1] 0.1

Here, we beat the salary only model along with the prevalence with a misclassification rate of 0.1. Our specificity looks great for our forward BIC model at 0.972. Our 7 false negatives could come down, however, with an adjustment to our cutoff value.

Find an Optimal Cutoff

Next we will define a function that will loop through a vector of potential cutoffs to isolate one that will produce the smallest misclassification rate with a minimal differential between sensitivity and specificity (see Appendix C for the opt_logistic_cutoff function’s definition).

(opt_cutoff = opt_logistic_cutoff(bic_model, cut_start = 0.1, cut_end = 0.8))

##      cutoff    misclass sensitivity specificity 
##  0.30000000  0.05555556  0.88888889  0.95833333
classify_and_diagnose(bic_model, cutoff = opt_cutoff["cutoff"])$confusion_matrix
##           actual
## prediction  N  Y
##          N 69  2
##          Y  3 16

Our routine finds an optimal cutoff value for our forward BIC model of 0.3. Our misclassification rate went down with our cutoff adjustment to 0.056. Additionally, our sensitivity went up significantly (0.889) and we only had a modest reduction in specificity (0.958). This model looks to be exceptionally adept at classifying division winners, and salary is not even a predictor!


Results

And the Winning Logistic Regression Model is…


Discussion


Apendices

Appendix A - Logistic Regression Models

Classify and Diagnose Function

classify_and_diagnose = function(model, data = bbproj_tst, 
                                 actual = bbproj_tst$DivWin, 
                                 pos = "Y", neg = "N", cutoff = 0.5) {
  # Generate a classifier given the model, data, cutoff, and positive and
  # negative responses
  pred = ifelse(predict(model, 
                        data, 
                        type = "response") > cutoff, pos, neg)
  
  # Generate the confusion matrix
  conf_mat = table(prediction = pred, actual = actual)

  # Calculate sensitivity, specificity, and the misclassification rate and
  # return them plus the confusion matrix
  list(confusion_matrix = conf_mat, 
       sensitivity = conf_mat[2, 2] / sum(conf_mat[, 2]), 
       specificity = conf_mat[1, 1] / sum(conf_mat[, 1]), 
       misclassification = mean(pred != actual))
}

Optimal Logistic Cutoff Function

opt_logistic_cutoff = function(model, cut_start = 0.01, cut_end = 0.99, 
                               data = bbproj_tst, actual = bbproj_tst$DivWin,
                               pos = "Y", neg = "N", plotit = TRUE) {
  # Loop through potential cutoffs from cut_start to cut_end to determine a 
  # cutoff that produces the lowest misclassification rate with the smallest 
  # delta between sensitivity and specificity
  cutoffs = seq(cut_start, cut_end, by = 0.01)
  sens = rep(0, length(cutoffs))
  spec = rep(0, length(cutoffs))
  misclass = rep(0, length(cutoffs))
  delta = rep(0, length(cutoffs))
  for (i in 1:length(cutoffs)) {
    diagnostics = classify_and_diagnose(model, 
                                        data = data, 
                                        actual = actual,
                                        pos = pos,
                                        neg = neg,
                                        cutoff = cutoffs[i])
    sens[i] = diagnostics$sensitivity
    spec[i] = diagnostics$specificity
    misclass[i] = diagnostics$misclassification
    delta[i] = abs(diagnostics$sensitivity - diagnostics$specificity)
  }
  
  # Get the indicies of the smallest misclassification rates
  min_misclass = which(misclass == min(misclass))
  
  # Get the smallest delta between sensitivity and specificity at the 
  # identified misclassification indicies
  min_delta = min_misclass[which.min(delta[min_misclass])]
  
  # Plot sensitivity, specificity, and misclassification if requested
  if (plotit) {
    plot(sens ~ cutoffs,
         xlab = "Cutoff",
         ylab = "Sensitivity/Specificity",
         main = "Sensitivity and Specificity at Varied Cutoffs",
         col = "tomato",
         type = "b",
         ylim = c(0, 1),
         pch = 20)
    lines(spec ~ cutoffs, 
          col = "darkslategray4", 
          type = "b",
          pch = 20)
    lines(misclass ~ cutoffs, 
          col = "darkorange", 
          type = "b",
          pch = 20)
    abline(v = cutoffs[min_delta], lty = 2)
    legend("topright", c("Sensitivity", 
                         "Specificity", 
                         "Misclassification"),
           col = c("tomato", "darkslategray4", "darkorange"),
           lwd = 1,
           pch = 20)
  }
  
  # Return the misclassification, sensitivity, and specificity at the optimal
  # cutoff value
  c(cutoff = cutoffs[min_delta],
    misclass = misclass[min_delta], 
    sensitivity = sens[min_delta],
    specificity = spec[min_delta])
}

Backward Search: BIC Model

formula <- formula(scope)
back_bic_model <- step(glm(formula, data = bbproj_trn, family = binomial), 
                       direction = "backward", k = log(n), trace = 0)
back_bic_diag <- classify_and_diagnose(back_bic_model)
summary(back_bic_model)
## 
## Call:
## glm(formula = DivWin ~ W + AB + H + HR + BB + HBP + IBB + SLG + 
##     OBP + wOBA, family = binomial, data = bbproj_trn)
## 
## Deviance Residuals: 
##      Min        1Q    Median        3Q       Max  
## -2.09816  -0.27642  -0.04275  -0.00332   2.76522  
## 
## Coefficients:
##                Estimate  Std. Error z value           Pr(>|z|)    
## (Intercept) -1052.50217   319.70312  -3.292           0.000994 ***
## W               0.37031     0.04793   7.726 0.0000000000000111 ***
## AB              0.17509     0.05585   3.135           0.001717 ** 
## H              -0.52892     0.16383  -3.228           0.001245 ** 
## HR              0.09433     0.03603   2.618           0.008842 ** 
## BB             -0.34709     0.10338  -3.357           0.000787 ***
## HBP            -0.33911     0.09686  -3.501           0.000463 ***
## IBB            -0.30746     0.12140  -2.533           0.011320 *  
## SLG          1794.92000   702.65755   2.554           0.010635 *  
## OBP          6178.22128  2030.78091   3.042           0.002348 ** 
## wOBA        -5414.02198  2077.07626  -2.607           0.009146 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 420.34  on 419  degrees of freedom
## Residual deviance: 179.06  on 409  degrees of freedom
## AIC: 201.06
## 
## Number of Fisher Scoring iterations: 8
back_bic_diag
## $confusion_matrix
##           actual
## prediction  N  Y
##          N 70  7
##          Y  2 11
## 
## $sensitivity
## [1] 0.6111111
## 
## $specificity
## [1] 0.9722222
## 
## $misclassification
## [1] 0.1

Optimal Cutoff

opt_logistic_cutoff(back_bic_model, cut_start = 0.1, cut_end = 0.8)

##      cutoff    misclass sensitivity specificity 
##  0.26000000  0.05555556  0.83333333  0.97222222
opt_back_bic_diag$confusion_matrix
##           actual
## prediction  N  Y
##          N 70  3
##          Y  2 15

Forward Search: BIC Model

forw_bic_model <- step(start_model, direction = "forward", scope = scope, 
                       k = log(n), trace = 0)
forw_bic_diag <- classify_and_diagnose(forw_bic_model)
summary(forw_bic_model)
## 
## Call:
## glm(formula = DivWin ~ W + X2B, family = binomial, data = bbproj_trn)
## 
## Deviance Residuals: 
##      Min        1Q    Median        3Q       Max  
## -2.35987  -0.30139  -0.06081  -0.00867   2.59348  
## 
## Coefficients:
##               Estimate Std. Error z value          Pr(>|z|)    
## (Intercept) -27.523331   3.832156  -7.182 0.000000000000686 ***
## W             0.350030   0.042623   8.212           < 2e-16 ***
## X2B          -0.016344   0.006758  -2.418            0.0156 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 420.34  on 419  degrees of freedom
## Residual deviance: 194.55  on 417  degrees of freedom
## AIC: 200.55
## 
## Number of Fisher Scoring iterations: 7
forw_bic_diag
## $confusion_matrix
##           actual
## prediction  N  Y
##          N 70  7
##          Y  2 11
## 
## $sensitivity
## [1] 0.6111111
## 
## $specificity
## [1] 0.9722222
## 
## $misclassification
## [1] 0.1

Optimal Cutoff

opt_logistic_cutoff(forw_bic_model, cut_start = 0.1, cut_end = 0.8)

##      cutoff    misclass sensitivity specificity 
##  0.30000000  0.05555556  0.88888889  0.95833333
opt_forw_bic_diag$confusion_matrix
##           actual
## prediction  N  Y
##          N 69  2
##          Y  3 16

Stepwise Search: BIC Model

step_bic_model <- step(start_model, direction = "both", scope = scope, 
                       k = log(n), trace = 0)
step_bic_diag <- classify_and_diagnose(step_bic_model)
summary(step_bic_model)
## 
## Call:
## glm(formula = DivWin ~ W + X2B, family = binomial, data = bbproj_trn)
## 
## Deviance Residuals: 
##      Min        1Q    Median        3Q       Max  
## -2.35987  -0.30139  -0.06081  -0.00867   2.59348  
## 
## Coefficients:
##               Estimate Std. Error z value          Pr(>|z|)    
## (Intercept) -27.523331   3.832156  -7.182 0.000000000000686 ***
## W             0.350030   0.042623   8.212           < 2e-16 ***
## X2B          -0.016344   0.006758  -2.418            0.0156 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 420.34  on 419  degrees of freedom
## Residual deviance: 194.55  on 417  degrees of freedom
## AIC: 200.55
## 
## Number of Fisher Scoring iterations: 7
step_bic_diag
## $confusion_matrix
##           actual
## prediction  N  Y
##          N 70  7
##          Y  2 11
## 
## $sensitivity
## [1] 0.6111111
## 
## $specificity
## [1] 0.9722222
## 
## $misclassification
## [1] 0.1

Optimal Cutoff

opt_step_bic <- opt_logistic_cutoff(step_bic_model, 
                                    cut_start = 0.1, 
                                    cut_end = 0.8,
                                    plotit = FALSE)
opt_step_bic_diag <- classify_and_diagnose(step_bic_model, 
                                           cutoff = opt_step_bic["cutoff"])
opt_logistic_cutoff(step_bic_model, cut_start = 0.1, cut_end = 0.8)

##      cutoff    misclass sensitivity specificity 
##  0.30000000  0.05555556  0.88888889  0.95833333
opt_step_bic_diag$confusion_matrix
##           actual
## prediction  N  Y
##          N 69  2
##          Y  3 16

Backward Search: AIC Model

formula <- formula(scope)
back_aic_model <- step(glm(formula, data = bbproj_trn, family = binomial), 
                       direction = "backward", trace = 0)
back_aic_diag <- classify_and_diagnose(back_aic_model)
summary(back_aic_model)
## 
## Call:
## glm(formula = DivWin ~ W + R + AB + H + HR + BB + CS + HBP + 
##     ER + IPouts + E + FP + BPF + PPF + RBI + IBB + SLG + OBP + 
##     OPS + wOBA, family = binomial, data = bbproj_trn)
## 
## Deviance Residuals: 
##      Min        1Q    Median        3Q       Max  
## -1.96822  -0.20843  -0.02111  -0.00065   2.65771  
## 
## Coefficients:
##                      Estimate        Std. Error z value      Pr(>|z|)    
## (Intercept)       -140.179255        693.655494  -0.202      0.839848    
## W                    0.545656          0.090798   6.010 0.00000000186 ***
## R                   -0.076857          0.035782  -2.148      0.031717 *  
## AB                   0.183520          0.063041   2.911      0.003601 ** 
## H                   -0.592491          0.183941  -3.221      0.001277 ** 
## HR                   0.079762          0.041066   1.942      0.052099 .  
## BB                  -0.401042          0.116424  -3.445      0.000572 ***
## CS                  -0.059004          0.023336  -2.528      0.011457 *  
## HBP                 -0.393193          0.110648  -3.554      0.000380 ***
## ER                   0.024676          0.008386   2.943      0.003255 ** 
## IPouts               0.022175          0.011255   1.970      0.048806 *  
## E                   -0.168360          0.092043  -1.829      0.067378 .  
## FP               -1111.221700        583.517366  -1.904      0.056865 .  
## BPF                  0.736250          0.339446   2.169      0.030084 *  
## PPF                 -0.674039          0.343047  -1.965      0.049431 *  
## RBI                  0.070973          0.036748   1.931      0.053438 .  
## IBB                 -0.299451          0.134303  -2.230      0.025769 *  
## SLG          851959707.701375  417793051.690715   2.039      0.041431 *  
## OBP          851964540.159953  417793092.777287   2.039      0.041430 *  
## OPS         -851957944.218280  417793035.107366  -2.039      0.041431 *  
## wOBA             -5332.125383       2319.746099  -2.299      0.021529 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 420.34  on 419  degrees of freedom
## Residual deviance: 151.22  on 399  degrees of freedom
## AIC: 193.22
## 
## Number of Fisher Scoring iterations: 8
back_aic_diag <- classify_and_diagnose(back_aic_model)

Optimal Cutoff

opt_back_aic <- opt_logistic_cutoff(back_aic_model, 
                                    cut_start = 0.1, 
                                    cut_end = 0.8,
                                    plotit = FALSE)
opt_back_aic_diag <- classify_and_diagnose(back_aic_model, 
                                           cutoff = opt_back_aic["cutoff"])
opt_logistic_cutoff(back_aic_model, cut_start = 0.1, cut_end = 0.8)

##      cutoff    misclass sensitivity specificity 
##  0.26000000  0.05555556  0.88888889  0.95833333
opt_back_aic_diag$confusion_matrix
##           actual
## prediction  N  Y
##          N 69  2
##          Y  3 16

Forward Search: AIC Model

forw_aic_model <- step(start_model, direction = "forward", 
                       scope = scope, trace = 0)
forw_aic_diag <- classify_and_diagnose(forw_aic_model)
summary(forw_aic_model)
## 
## Call:
## glm(formula = DivWin ~ W + X2B + HA + DP + GIDP, family = binomial, 
##     data = bbproj_trn)
## 
## Deviance Residuals: 
##      Min        1Q    Median        3Q       Max  
## -2.31485  -0.25596  -0.04893  -0.00565   2.90172  
## 
## Coefficients:
##               Estimate Std. Error z value Pr(>|z|)    
## (Intercept) -37.894889   6.186389  -6.126 9.04e-10 ***
## W             0.392678   0.048963   8.020 1.06e-15 ***
## X2B          -0.024866   0.007752  -3.208  0.00134 ** 
## HA            0.006988   0.002855   2.447  0.01440 *  
## DP           -0.024295   0.012063  -2.014  0.04401 *  
## GIDP          0.021012   0.012242   1.716  0.08609 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 420.34  on 419  degrees of freedom
## Residual deviance: 182.92  on 414  degrees of freedom
## AIC: 194.92
## 
## Number of Fisher Scoring iterations: 7
forw_aic_diag
## $confusion_matrix
##           actual
## prediction  N  Y
##          N 70  8
##          Y  2 10
## 
## $sensitivity
## [1] 0.5555556
## 
## $specificity
## [1] 0.9722222
## 
## $misclassification
## [1] 0.1111111

Optimal Cutoff

opt_forw_aic <- opt_logistic_cutoff(forw_aic_model, 
                                    cut_start = 0.1, 
                                    cut_end = 0.8,
                                    plotit = FALSE)
opt_forw_aic_diag <- classify_and_diagnose(forw_aic_model, 
                                           cutoff = opt_forw_aic["cutoff"])
opt_logistic_cutoff(forw_aic_model, cut_start = 0.1, cut_end = 0.8)

##      cutoff    misclass sensitivity specificity 
##  0.23000000  0.07777778  0.94444444  0.91666667
opt_forw_aic_diag$confusion_matrix
##           actual
## prediction  N  Y
##          N 66  1
##          Y  6 17

Stepwise Search: AIC Model

step_aic_model <- step(start_model, direction = "both", 
                       scope = scope, trace = 0)
step_aic_diag <- classify_and_diagnose(step_aic_model)
summary(step_aic_model)
## 
## Call:
## glm(formula = DivWin ~ W + X2B + HA + DP + GIDP, family = binomial, 
##     data = bbproj_trn)
## 
## Deviance Residuals: 
##      Min        1Q    Median        3Q       Max  
## -2.31485  -0.25596  -0.04893  -0.00565   2.90172  
## 
## Coefficients:
##               Estimate Std. Error z value Pr(>|z|)    
## (Intercept) -37.894889   6.186389  -6.126 9.04e-10 ***
## W             0.392678   0.048963   8.020 1.06e-15 ***
## X2B          -0.024866   0.007752  -3.208  0.00134 ** 
## HA            0.006988   0.002855   2.447  0.01440 *  
## DP           -0.024295   0.012063  -2.014  0.04401 *  
## GIDP          0.021012   0.012242   1.716  0.08609 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 420.34  on 419  degrees of freedom
## Residual deviance: 182.92  on 414  degrees of freedom
## AIC: 194.92
## 
## Number of Fisher Scoring iterations: 7
step_aic_diag
## $confusion_matrix
##           actual
## prediction  N  Y
##          N 70  8
##          Y  2 10
## 
## $sensitivity
## [1] 0.5555556
## 
## $specificity
## [1] 0.9722222
## 
## $misclassification
## [1] 0.1111111

Optimal Cutoff

opt_step_aic <- opt_logistic_cutoff(step_aic_model, 
                                    cut_start = 0.1, 
                                    cut_end = 0.8,
                                    plotit = FALSE)
opt_step_aic_diag <- classify_and_diagnose(step_aic_model, 
                                           cutoff = opt_step_aic["cutoff"])
opt_logistic_cutoff(step_aic_model, cut_start = 0.1, cut_end = 0.8)

##      cutoff    misclass sensitivity specificity 
##  0.23000000  0.07777778  0.94444444  0.91666667
opt_step_aic_diag$confusion_matrix
##           actual
## prediction  N  Y
##          N 66  1
##          Y  6 17